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Question Bank [2022] — Maths (commerce) STD 12 Commerce / Arts — Question

Maharashtra BoardEnglish MediumSTD 12 Commerce / ArtsMaths (commerce)Question Bank [2022]4 Marks
Question
A chemist has a compound to be made using 3 basic elements X, Y, Z so that it has at least 10 litres of X, 12 litres of Y and 20 litres of Z. He makes this compound by mixing two compounds (I) and (II). Each unit compound (I) had 4 litres of X, 3 litres of Y. Each unit compound (II) had 1 litre of X, 2 litres of Y and 4 litres of Z. The unit costs of compounds (I) and (II) are ₹ 400 and ₹ 600 respectively. Find the number of units of each compound to be produced so as to minimize the cost
✓

Answer

Let the chemist produce x units of compound I and y units of compound II.
Since x and y cannot be negative, x ≥ 0, y ≥ 0
The unit costs of compounds I and II are ₹ 400 and ₹ 600 respectively.
Total Cost = Z = 400x + 600y
We construct a table with constraints of X, Y and Z as follows:
  Compound I Compound II Least value
X 4 1 10
Y 3 2 12
Z – 4 20
From the table, the constraints are
4x + y ≥ 10
3x + 2y ≥ 12
4y ≥ 20
∴ Given problem can be formulated as follows:
Minimize Z = 400x + 600y
Subject to 4x + y ≥ 10
3x + 2y ≥ 12
4y ≥ 20, x ≥ 0, y ≥ 0
To draw the feasible region, construct table as follows:
Inequality $4 x+y \geq 10$ $3 x+2 y \geq 12$ $4 y \geq 20$
Corresponding equation (of line) 4x + y = 10 3x + 2y = 12 4y = 20
Intersection of line with X-axis $\left(\frac{5}{2}, 0\right)$ (4, 0) –
Intersection of line with Y-axis (0, 10) (0, 6) (0, 5)
Region Non-Origin side Non-Origin side Non-Origin side
Shaded portion EABY is the feasible region, whose vertices are A and B(0, 10).
A is the point of intersection of the lines 4y = 20 and 4x + y = 10,
Solving the above equations, we get
$x=\frac{5}{4}, y=5$
$\therefore A \equiv\left(\frac{5}{4}, 5\right)$

Here, the objective function is
Z = 400x + 600y
$ \therefore \text { Z at } A\left(\frac{5}{4}, 5\right)=400\left(\frac{5}{4}\right)+600(5)$
$=500+3000$
$=3500$
$Z \text { at } B(0,10)=400(0)+600(10)$
$=6000 $
$\therefore Z$ has minimum value 3500 at $x=\frac{5}{4}$ and $y=5$
$\therefore$ The chemist should produce $\frac{5}{4}$ units of compound I and 5 units of compound II to minimize the cost.

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